The Cylon Transponder is a small, round device that was found installed discretely under Galactica's DRADIS console. A similar device was carried about by the Humano-Cylon agent, Number Six, during her time on Caprica with Gaius Baltar (Mini-Series), perhaps to better allow tracking of herself by other Cylon agents.
The transponder works as an IFF device, identifying anyone or anything that holds it as a "friendly" contact to other Cylons. Apparently this device was installed by the first copy of Aaron Doral shortly before the Cylon attack began while security throughout the ship was more lax during the work of transforming Galactica into a museum. It was likely intended by the Cylons to track Galactica should she escape a Cylon attack.
Commander William Adama took strategic advantage of one of the devices by placing it aboard a Raptor, allowing it to pass unchallenged into a Cylon Basestar, where a nuclear device was placed aboard the Basestar to destroy it (Kobol's Last Gleaming, Part II). A second transponder was also in place on the captured Raider, piloted by Kara Thrace in her against-orders adventure back to Caprica in search of an ancient artifact (Kobol's Last Gleaming, Part I). It had been obtained from the copy of Leoben Conoy found on the Gemenon Traveller, in a scene cut from the episode "Flesh and Bone".
It is presumed that the transponder uses a form of wireless communication, although this has not been established in the series.
Keen eyes who watch the Mini-Series will note that the transponder was not on the DRADIS console until later in the show, indicating that it was placed either shortly before or after the ship entered combat.